Karl Pearson 
265 
(8) Numerical Illustrations. I propose in this section to illustrate the main 
results of the above investigation by selecting possible values for some of the 
constants. I am far from asserting that these are actually existing values, but 
they appear not improbable values — say, for the case of ability. We are only at 
present acquiring knowledge of the relationship between ability and fertility in 
the case of man, and the examples I give are merely illustrative and intended to 
indicate how the algebraical formulae can be dealt with. At the same time, I 
think, they throw some light on rather urgent national problems. It is desirable 
to bring home to the minds of the thinking classes what it really does mean, if the 
fittest in any character have not a third of the fertility of the least fit. 
Illustration I. The upper decile of a population has an average fertility rate 
of 2, the 50 "/^ showing the lower values of the character an average fertility rate 
of 6 The fertility of the whole population has a mean value of 5. Find the 
changes in the population during one generation. 
Probability integral tables give at once Jh = l"2816crj;. 
To determine Hi, we have 
Clearly h^ = 0, HkJ hJN xf = x Qjh = -Q, from which the tables give us 
= -2533. 
A rough diagram indicates at once the sign of the quantities and shows 
that we must have 
therefore 
X fj,JN X /= -1 X 2/5 = -04, 
1-7500. 
or in this case 
•2533o-^o-„/\/ o-o- + <j^' = <Tx'kl{a,r + a^') ; 
or, subtracting : 
will now give us y^. On substituting the values of and k and 5 for/ we find 
y, = 6-439. 
The distribution of fertility is accordingly given by 
_ 1 /a : - -SlSlo-A ^ 
y = G-439e 'A ^esTVTj _ 
